Multi-Irreducible Spectral Synchronization for Robust Rotation Averaging
CoRR(2023)
摘要
Rotation averaging (RA) is a fundamental problem in robotics and computer
vision. In RA, the goal is to estimate a set of $N$ unknown orientations
$R_{1}, ..., R_{N} \in SO(3)$, given noisy measurements $R_{ij} \sim R^{-1}_{i}
R_{j}$ of a subset of their pairwise relative rotations. This problem is both
nonconvex and NP-hard, and thus difficult to solve in the general case. We
apply harmonic analysis on compact groups to derive a (convex) spectral
relaxation constructed from truncated Fourier decompositions of the individual
summands appearing in the RA objective; we then recover an estimate of the RA
solution by computing a few extremal eigenpairs of this relaxation, and
(approximately) solving a consensus problem. Our approach affords several
notable advantages versus prior RA methods: it can be used in conjunction with
\emph{any} smooth loss function (including, but not limited to, robust
M-estimators), does not require any initialization, and is implemented using
only simple (and highly scalable) linear-algebraic computations and
parallelizable optimizations over band-limited functions of individual
rotational states. Moreover, under the (physically well-motivated) assumption
of multiplicative Langevin measurement noise, we derive explicit performance
guarantees for our spectral estimator (in the form of probabilistic tail bounds
on the estimation error) that are parameterized in terms of graph-theoretic
quantities of the underlying measurement network. By concretely linking
estimator performance with properties of the underlying measurement graph, our
results also indicate how to devise measurement networks that are
\emph{guaranteed} to achieve accurate estimation, enabling such downstream
tasks as sensor placement, network compression, and active sensing.
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